A generalisation of [101] (which asks about $k=4$).

The restriction to $k\geq 4$ is necessary since Sylvester has shown that $f_3(n)= n^2/6+O(n)$. (See also Burr, Gr\"{u}nbaum, and Sloane \cite{BGS74} and F\"{u}redi and Pal\'{a}sti \cite{FuPa84} for constructions which show that $f_3(n)\geq(1/6+o(1))n^2$.)

For $k\geq 4$, K\'{a}rteszi \cite{Ka63} proved\[f_k(n)\gg_k n\log n\](resolving a conjecture of Erd\H{o}s that $f_k(n)/n\to \infty$). Gr\"{u}nbaum \cite{Gr76} proved\[f_k(n) \gg_k n^{1+\frac{1}{k-2}}.\]Erd\H{o}s speculated this may be the correct order of magnitude, but Solymosi and Stojakovi\'{c} \cite{SoSt13} give a construction which shows\[f_k(n)\gg_k n^{2-O_k(1/\sqrt{\log n})}\]


References


[BGS74] Burr, Stefan A. and Gr\"{u}nbaum, Branko and Sloane, N. J. A., The orchard problem. Geometriae Dedicata (1974), 397-424.

[FuPa84] F\"{u}redi, Z. and Pal\'{a}sti, I., Arrangements of lines with a large number of triangles. Proc. Amer. Math. Soc. (1984), 561-566.

[Gr76] Gr\"{u}nbaum, Branko, New views on some old questions of combinatorial geometry. Colloquio Internazionale sulle Teorie Combinatorie
(Roma, 1973), Tomo I (1976), 451-468.

[Ka63] F. K\'{a}rteszi, Sylvester egy t\'{e}tel\'{e}r\H{o}l \'{e}s Erd\H{o}s egy sejt\'{e}s\'{e}r\H{o}l. Matematikai Lapok (1963), 3-10.

[SoSt13] Solymosi, J\'{o}zsef and Stojakovi\'C, Milo\vS, Many collinear {$k$}-tuples with no {$k+1$} collinear points. Discrete Comput. Geom. (2013), 811-820.